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Injective (Δ + 1)-coloring of planar graphs with girth 6

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Abstract

A vertex coloring of a graph G is called injective if every two vertices joined by a path of length 2 get different colors. The minimum number χ i (G) of the colors required for an injective coloring of a graph G is clearly not less than the maximum degree Δ(G) of G. There exist planar graphs with girth g ≥ 6 and χ i = Δ+1 for any Δ ≥ 2. We prove that every planar graph with Δ ≥ 18 and g ≥ 6 has χ i ≤ Δ + 1.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin

  2. Institute of Mathematics at Yakutsk State University, Yakutsk, Russia

    A. O. Ivanova

Authors
  1. O. V. Borodin
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  2. A. O. Ivanova
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Corresponding author

Correspondence to O. V. Borodin.

Additional information

Original Russian Text Copyright © 2011 Borodin O. V. and Ivanova A. O.

The authors were supported by the Russian Foundation for Basic Research (Grants 09-01-00244 and 08-01-00673). The second author was also supported by a grant of the President of the Russian Federation for Junior Scientists (Grant MK-2302.2008.1).

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Novosibirsk; Yakutsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 1, pp. 30–38, January–February, 2011.

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Borodin, O.V., Ivanova, A.O. Injective (Δ + 1)-coloring of planar graphs with girth 6. Sib Math J 52, 23–29 (2011). https://doi.org/10.1134/S0037446606010034

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  • DOI: https://doi.org/10.1134/S0037446606010034

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